All Questions

In Gouvea book $p$-adic numbers, on can find this corollary (7.4.11) Let $f(X) = 1+a_1X+a_2X^2+a_3X^3+\cdots$ be a power series which converges on the closed ball of radius $c = p^m$. Let $m_1, m_2, \...

Let $f(z)=\sum_{n\ge1}a_nz^n$ be a power series of $\mathbb C_p[[z]]$ where the $a_n$ are such that $|a_n|=1$ for every positive integer $n$. Consider $z_0\in\mathbb C_p$ such that $|z_0|<1$. Can ...

It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and ...

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$ ? I guess $\overline{\mathbb{F}_p}((t))$ is not unramified over $\mathbb{F}_p((t))$ because $\overline{\mathbb{F}_p}((t))$ ...

Let $(K,|\cdot|)$ be a complete local field and $\mathcal{O}$ be its ring of integers. Let $C$ be a complete algebraic closure of $K$ and let $\mathfrak{m}:=\{x\in \mathcal{O}_{C}~|~|x|<1\}$ where $...

Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...